A271951 Decimal expansion of (1/2) Product_{p prime} 1+1/(p-1)^3, a constant related to I. M. Vinogradov's proof of the "ternary" Goldbach conjecture.

1, 1, 5, 0, 4, 8, 0, 7, 7, 2, 3, 5, 6, 6, 1, 8, 5, 2, 7, 2, 7, 8, 4, 8, 8, 0, 7, 4, 3, 7, 4, 6, 9, 8, 0, 9, 0, 6, 3, 0, 3, 9, 3, 2, 9, 8, 5, 1, 1, 0, 8, 3, 6, 8, 0, 6, 8, 8, 1, 9, 3, 0, 5, 9, 0, 2, 2, 8, 2, 6, 3, 2, 3, 2, 5, 4, 3, 8, 0, 1, 3, 7, 1, 5, 7, 4, 0, 5, 2, 0, 9, 2, 9, 9, 4, 3, 6, 3, 8

Offset: 1

Author

Jean-François Alcover, Apr 17 2016

Keywords

• nonn
• cons

Examples

			1.150480772356618527278488074374698090630393298511083680688193059...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood Constants, p. 88.

Links

• Eric Weisstein's MathWorld, Goldbach Conjecture.
• Eric Weisstein's MathWorld, Vinogradov's theorem.
• Wikipedia, Goldbach's conjecture.
• Wikipedia, Vinogradov's theorem.

Crossrefs

Cf. A005597.

Programs

Mathematica

$MaxExtraPrecision = 1600; digits = 99; terms = 1600; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0, 0}, LinearRecurrence[{4, -6, 3}, {3, 12, 30}, terms + 10]]; r[n_Integer] := LR[[n]]; (1/2) Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First

PARI

(1/2) * prodeulerrat(1+1/(p-1)^3) \\ Amiram Eldar, Mar 14 2021